The routine may be called by the names f07hbf, nagf_lapacklin_dpbsvx or its LAPACK name dpbsvx.
f07hbf performs the following steps:
1.If , real diagonal scaling factors, , are computed to equilibrate the system:
Whether or not the system will be equilibrated depends on the scaling of the matrix , but if equilibration is used, is overwritten by and .
2.If or , the Cholesky decomposition is used to factor the matrix (after equilibration if ) as if or if , where is an upper triangular matrix and is a lower triangular matrix.
3.If the leading principal minor of is not positive definite, then the routine returns with . Otherwise, the factored form of is used to estimate the condition number of the matrix . If the reciprocal of the condition number is less than machine precision, is returned as a warning, but the routine still goes on to solve for and compute error bounds as described below.
4.The system of equations is solved for using the factored form of .
5.Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
6.If equilibration was used, the matrix is premultiplied by so that it solves the original system before equilibration.
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
1: – Character(1)Input
On entry: specifies whether or not the factorized form of the matrix is supplied on entry, and if not, whether the matrix should be equilibrated before it is factorized.
afb contains the factorized form of . If , the matrix has been equilibrated with scaling factors given by s. ab and afb will not be modified.
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hbf is called.
8: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array afb
must be at least
On entry: if , afb contains the triangular factor or from the Cholesky factorization or of the band matrix , in the same storage format as . If , afb is the factorized form of the equilibrated matrix .
On exit: if , afb returns the triangular factor or from the Cholesky factorization or .
If , afb returns the triangular factor or from the Cholesky factorization or of the equilibrated matrix (see the description of ab for the form of the equilibrated matrix).
9: – IntegerInput
On entry: the first dimension of the array afb as declared in the (sub)program from which f07hbf is called.
On entry: the first dimension of the array b as declared in the (sub)program from which f07hbf is called.
14: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array x
must be at least
On exit: if or , the solution matrix to the original system of equations. Note that the arrays and are modified on exit if , and the solution to the equilibrated system is .
15: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07hbf is called.
16: – Real (Kind=nag_wp)Output
On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix (after equilibration if that is performed), computed as .
17: – Real (Kind=nag_wp) arrayOutput
On exit: if or , an estimate of the forward error bound for each computed solution vector, such that where is the th column of the computed solution returned in the array x and is the corresponding column of the exact solution . The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
18: – Real (Kind=nag_wp) arrayOutput
On exit: if or , an estimate of the component-wise relative backward error of each computed solution vector (i.e., the smallest relative change in any element of or that makes an exact solution).
19: – Real (Kind=nag_wp) arrayWorkspace
20: – Integer arrayWorkspace
21: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The leading minor of order of is not positive definite, so the factorization could not be completed, and the solution has not been computed. is returned.
(or ) is nonsingular, but rcond is less than
machine precision, meaning that the matrix is singular to working precision.
Nevertheless, the solution and error bounds are computed because there
are a number of situations where the computed solution can be more accurate
than the value of rcond would suggest.
For each right-hand side vector , the computed solution is the exact solution of a perturbed system of equations , where
if , ;
if , ,
is a modest linear function of , and is the machine precision. See Section 10.1 of Higham (2002) for further details.
If is the true solution, then the computed solution satisfies a forward error bound of the form
If is the th column of , then is returned in and a bound on is returned in . See Section 4.4 of Anderson et al. (1999) for further details.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07hbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07hbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
When , the factorization of requires approximately floating-point operations, where is the number of superdiagonals.
For each right-hand side, computation of the backward error involves a minimum of floating-point operations. Each step of iterative refinement involves an additional operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form ; the number is usually or and never more than . Each solution involves approximately operations.